I got off work this evening from my second job and perused an old notebook as I sipped an Anchor Steam, waiting for the moment when I would shuffle off to sleep and wake up in a few hours and move across town, when my brain started racing after looking at some notes I'd made waiting in an airport in May of 2009.
After an hour and a half that simply rushed by, I'd come up with an iff math theorem about square numbers, and the easy half of the proof. Don't be confused, "iff" is not a typo, it stands for "if, and only if" and is the strongest type of theorem. Theorems are statements like "if P, then Q", but that doesn't necessarily imply that "if Q, then P". Take squares..."If A is a square, then A has four sides" is true (in normal Euclidean space, etc), but it doesn't necessarily hold that "if A has four sides, then A is a square". Rectangles and trapezoids and parallelograms all have four sides but don't have to be squares.
An "iff" theorem is one that is super strong and implies that "if P, then Q", as well as "if Q, then P". I was messing around, and I need to get to bed, but I'll share the gist of it: a and b are consecutive triangular numbers if, and only if, their sum, c, is a square number.
What this means is, if you take two consecutive triangular numbers and add them up you get a square number, and, on the other hand, any square number can be expressed as a sum of consecutive triangular numbers.
Crazy night...we move in a few hours!
not surprising you lost me there...but I did read and try to learn... I like the new picture where is it? Very nice morning fog nestled in the valleys. Hope you've recovered from your move.
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